A Claimed Zero-Free Region for Dirichlet L-Functions Begins at 11/12
An OpenAI preprint claims that every Dirichlet L-function, including the Riemann zeta function, has no zeros strictly to the right of Re s = 11/12—a fixed boundary that falls short of the Riemann hypothesis. Its proof, as presented in the manuscript, first bounds cancellation in a weighted sum over ideals in the Eisenstein integers, then uses a Mellin transform to turn that bound into a zero-free region. The key step is a deep mean-square estimate for a family of character sums; the paper’s account of that estimate is a roadmap rather than a full proof.
The claim is a fixed zero-free boundary, not the Riemann hypothesis
The manuscript claims that every Dirichlet L-function, including the Riemann zeta function, has no zeros strictly to the right of Re s = 11/12. Unlike a boundary that moves with the character’s conductor or with the height of a zero, this line is fixed.
That is a substantial but limited claim. It does not establish the Riemann hypothesis, which places the nontrivial zeros on Re s = 1/2; the region between 1/2 and 11/12 remains unresolved here. Nor does the claim address zeros exactly on the 11/12 boundary. The pole of zeta at s = 1 is not a zero and is not removed by the statement.
The proof’s route to this boundary is indirect: first establish cancellation in a weighted sum over ideals, then use that cancellation to rule out zeros.
A weighted sum over ideals is the object to control
The argument works in the number field Q(√−3), whose integers form the triangular lattice of Eisenstein integers. The sums are over ideals, with norm serving as a measure of size. For a fixed finite-order Hecke character ν, the manuscript considers an ideal Möbius sum weighted by ν and by a smooth function selecting norms around a scale D. Primes above 2 and 3, along with primes in the character’s conductor, are excluded.
The ideal Möbius function vanishes when a squared prime ideal divides its argument; otherwise its sign alternates with the number of prime factors. The resulting sum, A₁(D), has roughly D available terms. The aim is to show that cancellation reduces its size to at most D^(11/12+ε), for every positive ε.
Instead of estimating A₁(D) alone, the manuscript places it in a family. It inserts a sextic residue character χₙ(u), where u varies through Eisenstein integers. Away from common factors, these character values are sixth roots of unity; at u = 1, the original sum returns. The key analytic input is a mean-square estimate for this family, over rows with norm at most H = D^(1+θ), where 0 < θ ≤ 1/10. The bound is essentially D^(1+ε)H. This estimate is the deep step; the later exponent calculation uses it but does not prove it.
Sixth powers make many family members repeat the target
The local mechanism connecting the family back to A₁(D) is elementary. Choose a prime ideal with primary generator 𝔭, and set u = 𝔭⁶. If 𝔭 does not divide 𝔫, then χₙ(𝔭⁶) = 1, since a sixth root of unity raised to the sixth power is one. If 𝔭 divides 𝔫, the character is zero. Thus this family member agrees with the target sum except for terms divisible by 𝔭.
Choose prime norms between Y/2 and Y. The omitted terms contribute an error of order D/Y. There are roughly Y/log Y such primes, and setting Y = H^(1/6) puts their sixth powers inside the mean-square range. Each selected prime supplies a distinct row that approximates the same target.
Averaging over these rows turns the family estimate into a bound on A₁(D). The mean-square budget, divided across roughly Y/log Y rows, contributes D^(1+ε)H^(5/6) to the bound on |A₁(D)|²; the approximation errors contribute D²H^(−1/3). With H = D^(1+θ), the leading exponent before taking a square root is 11/6 + 5θ/6 + ε. After the square root it is 11/12 + 5θ/12 + ε, while the error term is smaller.
The positive θ cannot simply be set to zero. Instead, for any requested positive loss, choose θ sufficiently small first. The result is the claimed D^(11/12+ε) bound.
The mean-square estimate rests on a longer chain of analysis
The manuscript’s roadmap for the family estimate begins with Poisson summation in the row variable. Gauss sum identities combine the Möbius signs with sextic coefficients, producing cubic Gauss coefficients connected to Fourier coefficients of Kubota’s cubic theta function. To use the theta function’s transformation law, the sum is completed by adding cubed factors.
That completion creates a problem: a bound on the completed sum does not bound one of its parts, because the parts may cancel. The added cube terms therefore cannot just be deleted. The manuscript uses Möbius inversion: for a cube index b, the factor ∑ over d dividing b of μ(d) equals one when b = 1 and zero otherwise. This isolates the original part.
Small cube divisors are handled with the completed estimate. Large divisors require two further Poisson transformations and a transfer estimate. A positive gap between the row scale and the full target scale contracts the range at each step, allowing the argument to close after finitely many iterations.
For a square-free primary row coprime to 6, the key local calculation is −1 − 2 = −3 ≡ 3 mod 6. On residue classes coprime to the local prime, the transformed character is quadratic, making a quadratic large sieve available to control the completed mean square. This is a roadmap, not the full proof: fixed character twists, common factors, repeated prime factors and transformed weights require additional arguments.
Cancellation becomes a zero-free region through a Mellin transform
The final step turns the bound on A₁(D) into a constraint on L-functions. Take the Mellin transform of A₁(D) across scales D, with weight D^(−s). The cancellation bound makes this transform holomorphic for Re s > 11/12. For Re s > 1, direct integration identifies it with the Mellin transform of the smooth window divided by a Hecke L-function with the excluded Euler factors removed. Equivalently, the product of that L-function with the transform equals the Mellin transform of the window. Analytic continuation extends this product identity into the larger half-plane.
If a zero ρ lay strictly to the right of 11/12, choose the test window W(y) = y^(−ρ)φ(y), with φ nonnegative, nonzero and smoothly supported between 1 and 2. This choice is within the smooth compactly supported window class used in the argument. Its Mellin transform at ρ is ∫₁² φ(y) dy/y > 0. But the product identity, evaluated at the hypothetical zero, would force that transform to vanish. The contradiction excludes the zero.
The manuscript transfers the Hecke conclusion to Dirichlet L-functions by quadratic base change. The relation involves finite nonzero Euler factors, and the point s = 1 is handled separately. Intermediate constants may depend on the fixed character and window, while the claimed boundary does not. The companion 7/8 result is a different argument. The sixth-power step is elementary; the mean-square estimate and the analysis supporting it are the deeper parts of the proof.